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Semi-numerical evaluation of one-loop corrections

Description: We present a semi-numerical algorithm to calculate one-loop virtual corrections to scattering amplitudes. The divergences of the loop amplitudes are regulated using dimensional regularization. We treat in detail the case of amplitudes with up to five external legs and massless internal lines, although the method is more generally applicable. Tensor integrals are reduced to generalized scalar integrals, which in turn are reduced to a set of known basis integrals using recursion relations. The reduction algorithm is modified near exceptional configurations to ensure numerical stability. To test the procedure we apply these techniques to one-loop corrections to the Higgs to four quark process for which analytic results have recently become available.
Date: August 1, 2005
Creator: Ellis, R.K.; Giele, W.T.; Zanderighi, G. & /Fermilab
Partner: UNT Libraries Government Documents Department

Investigations of QCD at non-zero isospin density

Description: We investigate the QCD phase diagram as a function of isospin chemical potential at a fixed temperature by directly putting large numbers of {pi}{sup +}s into the system. Correlation functions of N {pi}{sup +}s systems involves N!N! contractions, and become extremely expensive when N is large. In order to alleviate this problem, a recursion relation of correlation functions has been derived in Ref. [1] that substantially reduces the number of independent contractions needed and makes the study of many pions systems be possible. In this proceeding this method is investigated numerically. We have also constructed a new method that is even more efficient, enabling us to study systems of up to 72 {pi}{sup +}s.
Date: December 1, 2011
Creator: Zhifeng Shi, William Detmold
Partner: UNT Libraries Government Documents Department

Bootstrapping One-Loop QCD Amplitudeswith General Helicities

Description: The recently developed on-shell bootstrap for computing one-loop amplitudes in non-supersymmetric theories such as QCD combines the unitarity method with loop-level on-shell recursion. For generic helicity configurations, the recursion relations may involve undetermined contributions from non-standard complex singularities or from large values of the shift parameter. Here we develop a strategy for sidestepping difficulties through use of pairs of recursion relations. To illustrate the strategy, we present sets of recursion relations needed for obtaining n-gluon amplitudes in QCD. We give a recursive solution for the one-loop n-gluon QCD amplitudes with three or four color-adjacent gluons of negative helicity and the remaining ones of positive helicity. We provide an explicit analytic formula for the QCD amplitude A{sub 6;1}(1{sup -}, 2{sup -}, 3{sup -}, 4{sup +}, 5{sup +}, 6{sup +}), as well as numerical results for A{sub 7;1}(1{sup -}, 2{sup -}, 3{sup -}, 4{sup +}, 5{sup +}, 6{sup +}, 7{sup +}), A{sub 8;1}(1{sup -}, 2{sup -}, 3{sup -}, 4{sup +}, 5{sup +}, 6{sup +}, 7{sup +}, 8{sup +}), and A{sub 8;1}(1{sup -}, 2{sup -}, 3{sup -}, 4{sup -}, 5{sup +}, 6{sup +}, 7{sup +}, 8{sup +}). We expect the on-shell bootstrap approach to have widespread applications to phenomenological studies at colliders.
Date: April 25, 2006
Creator: Berger, Carola F.; Bern, Zvi; Dixon, Lance J.; Forde, Darren & Kosower, David A.
Partner: UNT Libraries Government Documents Department

All One-loop Maximally Helicity Violating Gluonic Amplitudes in QCD

Description: We use on-shell recursion relations to compute analytically the one-loop corrections to maximally-helicity-violating n-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases, with helicities (-+-+++) and (-++-++), as well as numerical results for six, seven, and eight gluons. Our construction of the n-gluon amplitudes illustrates the relatively modest growth in complexity of the on-shell-recursive calculation as the number of external legs increases. These amplitudes add to the growing body of one-loop amplitudes known for all n, which are useful for studies of general properties of amplitudes, including their twistor-space structure.
Date: July 5, 2006
Creator: Berger, Carola F.; Bern, Zvi; Dixon, Lance J.; Forde, Darren & Kosower, David A.
Partner: UNT Libraries Government Documents Department

Constructing the S-matrix With Complex Factorization

Description: A remarkable connection between BCFW recursion relations and constraints on the S-matrix was made by Benincasa and Cachazo in 0705.4305, who noted that mutual consistency of different BCFW constructions of four-particle amplitudes generates nontrivial (but familiar) constraints on three-particle coupling constants - these include gauge invariance, the equivalence principle, and the lack of non-trivial couplings for spins > 2. These constraints can also be derived with weaker assumptions, by demanding the existence of four-point amplitudes that factorize properly in all unitarity limits with complex momenta. From this starting point, we show that the BCFW prescription can be interpreted as an algorithm for fully constructing a tree-level S-matrix, and that complex factorization of general BCFW amplitudes follows from the factorization of four-particle amplitudes. The allowed set of BCFW deformations is identified, formulated entirely as a statement on the three-particle sector, and using only complex factorization as a guide. Consequently, our analysis based on the physical consistency of the S-matrix is entirely independent of field theory. We analyze the case of pure Yang-Mills, and outline a proof for gravity. For Yang-Mills, we also show that the well-known scaling behavior of BCFW-deformed amplitudes at large z is a simple consequence of factorization. For gravity, factorization in certain channels requires asymptotic behavior {approx} 1/z{sup 2}.
Date: June 19, 2009
Creator: Schuster, Philip C.; /SLAC; Toro, Natalia & /Stanford U., ITP
Partner: UNT Libraries Government Documents Department

Bootstrapping One-Loop QCD Amplitudes

Description: We review the recently developed bootstrap method for the computation of high-multiplicity QCD amplitudes at one loop. We illustrate the general algorithm step by step with a six-point example. The method combines (generalized) unitarity with on-shell recursion relations to determine the not cut-constructible, rational terms of these amplitudes. Our bootstrap approach works for arbitrary configurations of gluon helicities and arbitrary numbers of external legs.
Date: September 8, 2006
Creator: Berger, Carola F.
Partner: UNT Libraries Government Documents Department

Cubature rules of prescribed merit

Description: We introduce a criterion for the evaluation of multidimensional quadrature, or cubature, rules for the hypercube: this is the merit of a rule, which is closely related to its trigonometric degree, and which reduces to the Zaremba figure of merit in the case of a lattice rule. We derive a family of rules Q{sub k}{sup a} having dimension s and merit 2{sup k}. These rules seem to be competitive with lattice rules with respect to the merit that can be achieved with a given number of abscissas.
Date: March 1996
Creator: Lyness, J. N. & Sloan, I. H.
Partner: UNT Libraries Government Documents Department

Chebyshev recursion methods: Kernel polynomials and maximum entropy

Description: The authors describe two Chebyshev recursion methods for calculations with very large sparse Hamiltonians, the kernel polynomial method (KPM) and the maximum entropy method (MEM). They are especially applicable to physical properties involving large numbers of eigenstates, which include densities of states, spectral functions, thermodynamics, total energies, as well as forces for molecular dynamics and Monte Carlo simulations. The authors apply Chebyshev methods to the electronic structure of Si, the thermodynamics of Heisenberg antiferromagnets, and a polaron problem.
Date: October 1, 1995
Creator: Silver, R.N.; Roeder, H.; Voter, A.F. & Kress, J.D.
Partner: UNT Libraries Government Documents Department

Exact invariants in resonance form for time-dependent potentials

Description: We have developed a framework for the momentum-resonance formulation of Lewis and Leach that casts new light into the nature of exact, explicitly time-dependent invariants for one-dimensional time-dependent potentials and produces additional examples of such invariants. The momentum-resonance formulation postulates that the invariant be a rational function of momentum with simple poles, which are called momentum resonances. We have shown that an invariant of resonance type can be written as a functional of the potential in terms of the solution of a system of linear algebraic equations; and we have obtained a single necessary and sufficient condition for a potential to admit an invariant of resonance type. These results were obtained by reformulating the problem in terms of a set of discrete moments that satisfy two separate recursion formulae. Invariants for new time-dependent potentials were obtained and previously known invariants were recovered.
Date: January 1, 1985
Creator: Lewis, H.R. & Goedert, J.
Partner: UNT Libraries Government Documents Department

Replica-space renormalization in random-field systems

Description: The critical behavior of random-field systems is characterized by exponentially long relaxation times. They may differ by orders of magnitude and slow modes have to be considered quenched with respect to the faster ones. This requires a drastic modification of the renormalization process in which the degrees of freedom are integrated out. A new replica-space renormalization procedure, to carry out the coarse-graining of the time intervals, is presented. We relate by recursion-relations the effective reduced dimensions of consecutive time scales. Their stable fixed-points yield the apparent dimensionalities for the longest and shortest time scales, which are insensitive to the exact behavior on intermediate scales. For Ising systems we find that d = 2 is the lower critical dimension in both regimes. The thermal exponents for d = 3, in the short-time observable regime, are related to those of the pure system in d = 2. This is consistent with the observations in field-cooled random antiferromagnets of the correlation length exponent nu approx. = 1 (by neutron scattering) and of the symmetric logarithmic divergence in the specific-heat (by linear birefringence). 17 refs., 3 figs.
Date: January 1, 1985
Creator: Shapir, Y.
Partner: UNT Libraries Government Documents Department

Transition of fractal dimension in a latticed dynamical system

Description: We study a recursion relation that manifests two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigenbaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns are reproduced in this new route. In the oscillatory instability regime the fracal dimension saturates at D/sub F/ approx. = 2.6 with imbedding dimensions while in the turbulent regime D/sub F/ saturates at 6.0. 19 refs., 3 figs.
Date: March 1, 1986
Creator: Duong-van, M.
Partner: UNT Libraries Government Documents Department

Limit theorem for the maximum of an exponential autoregressive process. Technical report No. 14

Description: The asymptotic behavior of the maximum of a particular autoregressive process is discussed. The process was introduced by Gaver and Lewis in 1975 as a generalization of the Poisson process which allows for some dependence in the successive interarrival times. The exact distribution of the maximum of the first two terms and the first three terms in the sequence (denoted by M/sub 1/ and M/sub 2/, respectively) is calculated, and upper and lower bounds are obtained for M/sub n/, the maximum of the first n + 1 terms in the sequence. Loynes in 1965 gave conditions under which a stationary stochastic process has a maximum which behaves in the limit just as for independent identically distributed variables with the marginal distribution of the stationary process. Such conditions are often very difficult to verify in practice. However, for this particular example the joint distribution of the process at time n and n + j + 1 is determined, and the conditions to obtain the limit theorem are verified by use of the Markov property.
Date: September 16, 1977
Creator: Chernick, M
Partner: UNT Libraries Government Documents Department

Stochastic propagation of an array of parallel cracks: Exploratory work on matrix fatigue damage in composite laminates

Description: Transverse cracking of polymeric matrix materials is an important fatigue damage mechanism in continuous-fiber composite laminates. The propagation of an array of these cracks is a stochastic problem usually treated by Monte Carlo methods. However, this exploratory work proposes an alternative approach wherein the Monte Carlo method is replaced by a more closed-form recursion relation based on fractional Brownian motion.'' A fractal scaling equation is also proposed as a substitute for the more empirical Paris equation describing individual crack growth in this approach. Preliminary calculations indicate that the new recursion relation is capable of reproducing the primary features of transverse matrix fatigue cracking behavior. Although not yet fully tested or verified, this cursion relation may eventually be useful for real-time applications such as monitoring damage in aircraft structures.
Date: September 1, 1989
Creator: Williford, R.E.
Partner: UNT Libraries Government Documents Department

Bootstrapping Multi-Parton Loop Amplitudes in QCD

Description: The authors present a new method for computing complete one-loop amplitudes, including their rational parts, in non-supersymmetric gauge theory. This method merges the unitarity method with on-shell recursion relations. It systematizes a unitarity-factorization bootstrap approach previously applied by the authors to the one-loop amplitudes required for next-to-leading order QCD corrections to the processes e{sup +}e{sup -} {yields} Z, {gamma}* {yields} 4 jets and pp {yields} W + 2 jets. We illustrate the method by reproducing the one-loop color-ordered five-gluon helicity amplitudes in QCD that interfere with the tree amplitude, namely A{sub 5;1}(1{sup -}, 2{sup -}, 3{sup +}, 4{sup +}, 5{sup +}) and A{sub 5;1}(1{sup -}, 2{sup +}, 3{sup -}, 4{sup +}, 5{sup +}). Then we describe the construction of the six- and seven-gluon amplitudes with two adjacent negative-helicity gluons, A{sub 6;1}(1{sup -}, 2{sup -}, 3{sup +}, 4{sup +}, 5{sup +}, 6{sup +}) and A{sub 7;1}(1{sup -}, 2{sup -}, 3{sup +}, 4{sup +}, 5{sup +}, 6{sup +}, 7{sup +}), which uses the previously-computed logarithmic parts of the amplitudes as input. They present a compact expression for the six-gluon amplitude. No loop integrals are required to obtain the rational parts.
Date: July 6, 2005
Creator: Bern, Zvi; /UCLA; Dixon, Lance J.; /SLAC; Kosower, David A. & /Saclay, SPhT
Partner: UNT Libraries Government Documents Department

The Last of the Finite Loop Amplitudes in QCD

Description: We use on-shell recursion relations to determine the one-loop QCD scattering amplitudes with a massless external quark pair and an arbitrary number (n - 2) of positive-helicity gluons. These amplitudes are the last of the unknown infrared- and ultraviolet-finite loop amplitudes of QCD. The recursion relations are similar to ones applied at tree level, but contain new non-trivial features corresponding to poles present for complex momentum arguments but absent for real momenta. We present the relations and the compact solutions to them, valid for all n. We also present compact forms for the previously-computed one-loop n-gluon amplitudes with a single negative helicity and the rest positive helicity.
Date: May 31, 2005
Creator: Bern, Zvi; Dixon, Lance J. & Kosower, David A.
Partner: UNT Libraries Government Documents Department

Recursive Construction of Higgs-Plus-Multiparton Loop Amplitudes:The Last of the \phi-nite Loop Amplitudes

Description: We consider a scalar field, such as the Higgs boson H, coupled to gluons via the effective operator H tr G{sub {mu}{nu}} G{sup {mu}{nu}} induced by a heavy-quark loop. We treat H as the real part of a complex field {phi} which couples to the self-dual part of the gluon field-strength, via the operator {phi} tr G{sub SD {mu}{nu}} G{sub SD}{sup {mu}{nu}}, whereas the conjugate field {phi} couples to the anti-self-dual part. There are three infinite sequences of amplitudes coupling {phi} to quarks and gluons that vanish at tree level, and hence are finite at one loop, in the QCD coupling. Using on-shell recursion relations, we find compact expressions for these three sequences of amplitudes and discuss their analytic properties.
Date: August 18, 2006
Creator: Berger, Carola F.; Del Duca, Vittorio & Dixon, Lance J.
Partner: UNT Libraries Government Documents Department

Constructing QCD one-loop amplitudes

Description: In the context of constructing one-loop amplitudes using a unitarity bootstrap approach we discuss a general systematic procedure for obtaining the coefficients of the scalar bubble and triangle integral functions of one-loop amplitudes. Coefficients are extracted after examining the behavior of the cut integrand as the unconstrained parameters of a specifically chosen parameterization of the cut loop momentum approach infinity. Measurements of new physics at the forthcoming experimental program at CERN's Large Hadron Collider (LHC) will require a precise understanding of processes at next-to-leading order (NLO). This places increased demands for the computation of new one-loop amplitudes. This in turn has spurred recent developments towards improved calculational techniques. Direct calculations using Feynman diagrams are in general inefficient. Developments of more efficient techniques have usually centered around unitarity techniques [1], where tree amplitudes are effectively 'glued' together to form loops. The most straightforward application of this method, in which the cut loop momentum is in D = 4, allows for the computation of 'cut-constructible' terms only, i.e. (poly)logarithmic containing terms and any related constants. QCD amplitudes contain, in addition to such terms, rational pieces which cannot be derived using such cuts. These 'missing' rational parts can be extracted using cut loop momenta in D = 4-2 {var_epsilon}. The greater difficulty of such calculations has restricted the application of this approach, although recent developments [3, 4] have provided new promise for this technique. Recently the application of on-shell recursion relations [5] to obtaining the 'missing' rational parts of one-loop processes [6] has provided an alternative very promising solution to this problem. In combination with unitarity methods an 'on-shell bootstrap' approach provides an efficient technique for computing complete one-loop QCD amplitudes [7]. Additionally other new methods have also proved fruitful for calculating rational terms [8]. Such developments have again refocused attention on the ...
Date: February 22, 2008
Creator: Forde, Darren
Partner: UNT Libraries Government Documents Department

Recursive estimation for the tracking of radioactive sources

Description: This paper describes a recursive estimation algorithm used for tracking the physical location of radioactive sources in real-time as they are moved around in a facility. The algorithm is related to a nonlinear least squares estimation that minimizes the change in the source location and the deviation between measurements and model predictions simultaneously. The measurements used to estimate position consist of four count rates reported by four different gamma ray detectors. There is an uncertainty in the source location due to the large variance of the detected count rate. This work represents part of a suite of tools which will partially automate security and safety assessments, allow some assessments to be done remotely, and provide additional sensor modalities with which to make assessments.
Date: December 31, 1998
Creator: Howse, J.W.; Ticknor, L.O. & Muske, K.R.
Partner: UNT Libraries Government Documents Department

SATER/LLNL: an interactive identification/estimation/control package

Description: This issue of the computer program newsletter focuses on SATER/LLNL - an interactive code for identification, estimation, and classical control. A brief description of the SATER/LLNL operation is included as well as a sample session. Attached is the fundamental reference that details more of the package.
Date: September 1, 1982
Creator: Candy, J V & Journeay, C H
Partner: UNT Libraries Government Documents Department

Finite-particle-number approach to physics

Description: Starting from a discrete, self-generating and self-organizing, recursive model and self-consistent interpretive rules we construct: the scale constants of physics (3,10,137,1.7x10/sup 38/); 3+1 Minkowski space with a discrete metric and the algebraic bound ..delta.. is an element of ..delta.. tau is greater than or equal to 1; the Einstein-deBroglie relation; algebraic double slit interference; a single-time momentum-space scattering theory connected to laboratory experience; an approximation to wave functions; local phase severance and hence both distant correlations and separability; baryon number, lepton number, charge and helicity; m/sub p//m/sub e/; a cosmology not in disagreement with current observations.
Date: October 1, 1982
Creator: Noyes, H.P.
Partner: UNT Libraries Government Documents Department

A generalized model for coincidence counting

Description: The aim of this paper is to provide a description of the multiplicative processes associated with coincidence counting techniques, for example in the NDA of plutonium bearing materials. The model elucidates both the physical processes and the underlying mathematical formalism in a relatively simple but comprehensive way. In particular, it includes the effect of absorption by impurities or poisons, as well as that of neutron leakage on a parallel basis to the treatment of induced fission itself. The work thus parallels and generalizes the methods of Boehnel of Hage and Cifarelli, and more recently of Yanjushkin. This paper introduces the concept of a dual probability generating function to account for both the basic physical multiplication phenomena, as well as the detection phenomena. The underlying approach extends the idea of a simple probability generating function, due to De Moivre. The basic mathematical background may be found, for example, in Feller 1966.
Date: June 1, 1992
Creator: Lu, Ming-Shih & Teichmann, T.
Partner: UNT Libraries Government Documents Department